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A note on Kähler manifolds with almost nonnegative bisectional curvature

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Abstract

In this note, we prove the following result. There is a positive constant ε(n, Λ) such that if M n is a simply connected compact Kähler manifold with sectional curvature bounded from above by Λ, diameter bounded from above by 1, and with holomorphic bisectional curvature H ≥ −ε(n, Λ), then M n is diffeomorphic to the product M 1 × ⋯ × M k , where each M i is either a complex projective space or an irreducible Kähler–Hermitian symmetric space of rank ≥ 2. This resolves a conjecture of Fang under the additional upper bound restrictions on sectional curvature and diameter.

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References

  1. Bando S.: On the classification of three-dimensional compact Kaehler manifolds of nonnegative bisectional curvature. J. Diff. Geom 19(2), 283–297 (1984)

    MATH  MathSciNet  Google Scholar 

  2. Fang F.: Kähler manifolds with almost non-negative bisectional curvature. Asian J. Math. 6(3), 385–398 (2002)

    MATH  MathSciNet  Google Scholar 

  3. Fang F., Rong X.: Fixed point free circle actions and finiteness theorems. Comm. Contemp. Math. 2, 75–96 (2000)

    MATH  MathSciNet  Google Scholar 

  4. Hamilton R.: Three-manifolds with positive Ricci curvature. J. Diff. Geom. 17, 255–306 (1982)

    MATH  MathSciNet  Google Scholar 

  5. Hamilton R.: Four-manifolds with positive curvature operator. J. Diff. Geom. 24, 153–179 (1986)

    MATH  MathSciNet  Google Scholar 

  6. Hamilton R.: A compactness property for solutions of the Ricci flow. Amer. J. Math. 117(3), 545–572 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  7. Mok N.: The uniformization theorem for compact Kähler manifolds of non-negative holomorphic bisectional curvature. J. Diff. Geom. 27(2), 179–214 (1988)

    MATH  MathSciNet  Google Scholar 

  8. Petersen, P., Tao, T.: Classifiacation of almost quarter-pinched manifolds, to appear in Proc. Amer. Math. Soc. arXiv:0807.1900v1

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Authors and Affiliations

  1. School of Mathematical Sciences, Key Laboratory of Mathematics and Complex Systems, Beijing Normal University, Beijing, 100875, Peoples’ Republic of China

    Hong Huang

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  1. Hong Huang
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Correspondence to Hong Huang.

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Huang, H. A note on Kähler manifolds with almost nonnegative bisectional curvature. Ann Glob Anal Geom 36, 323–325 (2009). https://doi.org/10.1007/s10455-009-9165-9

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  • DOI: https://doi.org/10.1007/s10455-009-9165-9

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